Meditation upon this hierarchy in the context of 3-d turbulence produces some parallels. The distinction between wave turbulence and path-integral techniques invites a comparison between Eulerian [1]} and Lagrangian [2]} formulations of 3-D turbulence, with Eulerian formulations being prone to contamination by Doppler shifting. The wave kinetic equation represents nonlinear interactions as an amplitude modulation of spatially infinite plane waves and predicts a very rapid adjustment of a spike inserted into an otherwise smooth spectrum at a time scale \(\Gamma ^{-1}\) . At oceanic amplitudes, this adjustment time scale tends to an aphysical rms Doppler shift. Path-integral techniques essentially provide an average of that spectral spike with its interaction partners, and explicitly represents variations in Doppler shifting as the underpinning dynamics. In 3-D turbulence, the change from Eulerian to Lagrangian perspectives changes predicted power laws from \(k^{-3/2}\)  and \(k^{-5/3}\) . In the internal wave problem, the concept of ensemble averaging wave packets following ray trajectories provides motivation to include a mean drift term in the transport equation. This is a qualitative difference, and revised estimates of downscale transport overshoot the mark by an order of magnitude. These quantitative differences are potentially resolved by parallels concerning the role of fourth order cumulants. In 3-D turbulence, fourth order cumulants are understood to provide a systematic damping of downscale transports associated with third order terms [3]}. Here we propose a Bragg scattering process that reduces the downscale transports associated with a phase-velocity / group velocity resonance. We interpret Bragg scattering as a fourth order cumulant playing the role of a third order damping.
